One big thing that this does is mentioned in the problem: warming up. When you're under pressure, you don't want to just step off the street and start talking

One big thing that this does is mentioned in the problem: warming up. When you're under pressure, you don't want to just step off the street and start talking about some super-hard problem. But if you don't warm up at all, this is exactly what happens. And that's just the candidate! This also lets the interviewer get into interviewing mode. An interview is basically a type of performance. And what performer steps on stage without warming up?